91Ó°ÊÓ

When a capacitor is charged, it stores energy in an electric field between its plates. This energy can be calculated using the formula: \[ U = \frac{1}{2} C V^2 \]where \( U \) is the stored energy, \( C \) is the capacitance, and \( V \) is the voltage across the capacitor. In simple terms, this formula tells us that the energy depends on both how much charge the capacitor can hold (capacitance), and the potential difference between its plates (voltage). For example, using the given values from the exercise, a capacitor with a capacitance of 7.00 \( \mu \mathrm{F} \) and a voltage of 16.0 V can store a total energy of 0.000896 Joules after calculation.

• Capacitance (\( C \)) affects how much charge can be stored.
• Voltage (\( V \)) influences the amount of energy stored.

This stored energy is crucial because it can be later transferred to other components in the circuit, such as an inductor.

In an LC circuit, when an initially charged capacitor is connected to an inductor, the energy stored in the capacitor is gradually transferred to the inductor. At the moment when all this energy has shifted completely into the inductor, the current through the inductor reaches its maximum value.This maximum inductor current can be calculated using the equation for energy conservation:\[ U = \frac{1}{2} L I_{\text{max}}^2 \]Here, \( U \) is the same energy initially stored in the capacitor, \( L \) is the inductance, and \( I_{\text{max}} \) is the maximum current.By rearranging and solving, we find \( I_{\text{max}} \) to be approximately 0.691 A, using the inductance of 3.75 mH. The initial energy from the capacitor's electric field fully converts into magnetic energy in the inductor at this point.

• Maximum current is reached when all energy is transferred to the inductor.
• The formula relates energy with inductance and current.

In an LC circuit, the charge and current are dynamically changing due to the continuous energy exchange between the capacitor and the inductor. At the moment when the current through the inductor is at its maximum, the charge on the capacitor plates is zero. Why is this? Because all the energy stored in the capacitor has been converted to kinetic energy in the form of the current flowing through the inductor.Thus, the charge (\( q \)) on the capacitor is:\[ q = 0 \text{ C} \]at that instant. There are different stages in this energy oscillation, but at this specific moment, energy rests only in the inductor.

• Charge is zero when maximum current flows through the inductor.
• Energy forms a cycle between the capacitor and the inductor.